Why should I buy a ketch?

It will look like a proper sailing ship, it will give smaller more easily managed sails and give you the option of the classic ketch rig of mizen and jib in bad weather.
 
Much depends on the design, but, in general:

Windward performance will suffer significantly and there is a lot more expense involved in dealing with two masts and sets of rigging. More strings to pull, so a little more complicated to sail. Again, depending on design, you may find she is underpowered on the wind or on the run, because the mizzen may not work well on those points.

The mizzen mast may be a bit "in the way" in the cockpit in some boats. But not others.

On the plus side, sail combinations will be a lot more versatile. Jib and mizzen is a very easy going and effective combination in heavy weather (not too heavy - in a really big wind you need to bring sail inboard and use the mainmast, which is generally better stayed) A mizzen staysail is a powerful addition to your reaching wardrobe, as well as all the other usual sails. She will do very well on anything from a fine to a very broad reach.

She will be easy to balance on most points of sailing. You probably won't be able to use a wind vane, but you should be able to set her up to look after herself, using the mizzen. Sail areas will be easy to handle, so she will be easy on her crew. You won't often need an autopilot. You won't need such big winches or so much muscle power to handle her. With a well designed ketch, you can sit on the taffrail and steer her with the mizzen sheet.

She should heave to beautifully and in a squall you can drop the main at leisure, stay hove to under jib and mizzen and get under way with no fuss. If you get the combination right, she will ride steadily and quietly, even in big winds and seas.

The mizzen will hold you head to wind whilst you hoist or hand sail. It will do the same for you at anchor or in any other situation where you need to lie easy. It will also balance bow windage for you under power.

Oh, and you will look like a proper ship.

To sum up - if you want to race or eat up the wind, go for a sloop. If you want easy going sailing and passage making and don't worry too much about pointing ability, a ketch might have a lot going for it.
 
Don't forget the option of a mizzen staysail for broad reaching or boomed out for down wind. Smaller sails are easier to handle. I haven't noticed much loss of windward performance, but mine is a bilge keeler anyway & I cruise rather than race.

Comment above mentions added cost & complication of extra masts/ rigging, but masts are smaller and, in general less stressed as sails are also smaller. I still have my original masts & rigging after 35 years, but I am not a world-girdling master mariner, nor do I sail her on her ear all the time.

In short, I have had her 20 years and feel no need to change her yet.
 
We bought ours because we fell for the boat - ketch or sloop would have been immaterial. We'd certainly have another, but would not necessarily seek one out to the exclusion of considering any other rig.

As others have said, the mizzen is a great mounting for radar etc and avoids the problem of chafe on the genoa, it also provides (in our case) a real secure 'cage' effect which SWMBO loves - you'd have to try really hard to fall out of the cockpit. The mizzen provides a handy crane for lifting outboards and diving cylinders and despite the overhang, interference with the 'vane gear is not an issue. The mizzen staysail is a fantastic sail and so easily handle - ease the halyard and it virtually falls in the cockpit (and this from a sail slightly larger than our mainsail). On our particular class, I also believe that the ketch is prettier than the sloop (the original design was for a ketch). Rig options are increased (jib and mizzen is a really easy and surprisingly effective rig) and you have the makings of a really good jury rig and for that reason, I prefer not to have a triatic stay.

Downsides? The mizzen doesn't add a great deal upwind, increases drag and obviously re-rigging is more expensive. Don't be persuaded that ketches don't go upwind well though, treat individual boats on the'ir merits. Being a ketch didn't stop Steinlager wining the Whitbread!!!!
 
A key question might be; 'At what size does a ketch make sense?'
I've mostly sailed sloop-rigged boats, but a few ketches of different sizes, and I would say that a small one is a bit of nonesense.
(I await the backlash!)
On a small boat the extra rigging interferes with human progress; the mast is not stout enough to be a good mounting for equipment; the mizzen is likely to be too close to the leech of the main and back-winded by it (or the main has to be to be very high aspect-ratio, which negates the aim of a robust rig).
For my money, the ketch begins to make sense over 36ft LOA: below that, I prefer a sloop every time: over 40ft LOA I would go for a ketch, no question.
 
There are a number of sub 36ft ketches that are very credible, various Westerlies, Seadog, Barbary and Fisher to name a few. And going down in size even further, Drascombes which are far from "nonsense".
 
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